Question 1198690: There are 200 students in a particular graduate program at a state university. Of them, 110 are female and 125 are out- of state students. Of the 110 females, 70 are out - of - state students. Draw and label a Venn diagram depicting this information.
Are the events” female” and “ out - of - state student” independent? Are they mutually exclusive? Explain why or why not.
If one of these 200 students is selected at random, what is the probability that the student selected is an out - of - state student given this student is a female?
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Here's the Venn Diagram
I started with the 70 female out-of-state students in the overlapped region of the two circles.
Since there are 110 female students, there must be 110-70 = 40 female students who aren't out-of-state.
In other words, these female students are in-state.
This value goes inside the circle on the left, but outside the other circle.
There are 125 out-of-state students, which means 125-70 = 55 out-of-state students aren't female.
This value goes inside the circle on the right, but outside the other circle.
The values in either circle add to 40+70+55 = 165 to represent the number of students who are female, out-of-state, or both.
There are a total of 200 people surveyed, so we have 200-165 = 35 students who are neither female nor out-of-state.
We'll have this final value outside of both circles.
As a check, all four values should add to the grand total of 200.
The rectangle represents the universal set, which means every value is inside it.
Here is another way to represent those set of values in a two-way table.
| In-State | Out-Of-State | | Female | 40 | 70 | | Male | 35 | 55 |
and this is what it looks like when we have row and column totals
| In-State | Out-Of-State | Total | | Female | 40 | 70 | 110 | | Male | 35 | 55 | 90 | | Total | 75 | 125 | 200 |
Define these two events
A = person selected is female
B = person selected is out-of-state
From the venn diagram, and also the given info:
P(A) = 110/200 = 0.55
P(B) = 125/200 = 0.625
P(A and B) = 70/200 = 0.35
If events A and B were independent, then
P(A and B) = P(A)*P(B)
would be a true equation.
Let's see if that's the case
P(A)*P(B) = 0.55*0.625 = 0.34375
which is close to the 0.35 calculated earlier, but not quite there.
Therefore, P(A and B) = P(A)*P(B) is false in this case; meaning A and B are not independent.
Now on to see if the events are mutually exclusive.
A and B are mutually exclusive if and only if
P(A and B) = 0
but
P(A and B) = 0.35 which clearly is nonzero.
Events A and B are are not mutually exclusive
Now onto the last part of this question: If one of these 200 students is selected at random, what is the probability that the student selected is an out-of-state student given this student is a female?
The part at the end "given this student is a female" tells us that we focus on everyone inside the "female" circle of the venn diagram.
Ignore anyone else.
Of those 110 women, 70 are out-of-state
70/110 = 7/11 is the probability of selecting an out-of-state person if we know 100% the person is a woman.
7/11 = 0.63636 = 63.636% approximately
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Summary:
Question: Are the events "female" and "out-of-state student" independent?
Answer: No, the events are not independent.
Question: Are the events "female" and "out-of-state student" mutually exclusive?
Answer: No, the events are not mutually exclusive.
Question: If one of these 200 students is selected at random, what is the probability that the student selected is an out-of-state student given this student is a female?
Answer: 7/11
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